TOPOLOGY OPTIMIZATION BY THE HOMOGENIZATION - CMAP
Transcript of TOPOLOGY OPTIMIZATION BY THE HOMOGENIZATION - CMAP
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OPTIMAL DESIGN OF STRUCTURES (MAP 562)
G. ALLAIRE
February 11th, 2015
Department of Applied Mathematics, Ecole Polytechnique
CHAPTER VII (continued)
TOPOLOGY OPTIMIZATION
BY THE HOMOGENIZATION METHOD
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7.5 Shape optimization in the elasticity setting
Very similar to the conductivity setting but there are some additional hurdles.
We shall review the results without proofs.
The basic ingredients of the homogenization method are the sames:
introduction of composite designs characterized by (θ, A∗),
Hashin-Shtrikman bounds for composites,
sequential laminates are optimal microstructures.
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Model problem: compliance minimization
ΓD
N
Ω
Γ
D
Γ
Bounded working domain D ∈ IRN (N = 2, 3).
Linear isotropic elastic material, with Hooke’s law A
A = (κ− 2µ
N)I2 ⊗ I2 + 2µI4, 0 < κ, µ < +∞
Admissible shape = subset Ω ⊂ D.
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Boundary ∂Ω = Γ ∪ ΓN ∪ ΓD with ΓN and ΓD fixed.
− divσ = 0 in Ω
σ = 2µe(u) + λ tr(e(u)) Id in Ω
u = 0 on ΓD
σn = g on ΓN
σn = 0 on Γ,
Weight is minimized and rigidity is maximized. Let ℓ > 0 be a Lagrange
multiplier, the objective function is
infΩ⊂D
J(Ω) =
∫
ΓN
g · u ds+ ℓ
∫
Ω
dx
.
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This shape optimization problem can be approximated by a two-phase
optimization problem: the original material A and the holes of rigidity B ≈ 0.
The Hooke’s law of the mixture in D is
χΩ(x)A+(
1− χΩ(x))
B ≈ χΩ(x)A
The admissible set is
Uad =
χ ∈ L∞ (D; 0, 1)
.
As in conductivity/membrane case we can apply the relaxation approach
based on homogenization theory.
The homogenization method can be generalized to the elasticity setting.
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Homogenized formulation of shape optimization
We introduce composite structures characterized by a local volume fraction
θ(x) of the phase A (taking any values in the range [0, 1]) and an homogenized
tensor A∗(x), corresponding to its microstructure.
The set of admissible homogenized designs is
U∗ad =
(θ, A∗) ∈ L∞(
D; [0, 1]× IRN4)
, A∗(x) ∈ Gθ(x) in D
.
The homogenized state equation is
σ = A∗e(u) with e(u) = 12 (∇u+ (∇u)t) ,
divσ = 0 in D,
u = 0 on ΓD
σn = g on ΓN
σn = 0 on ∂D \ (ΓD ∪ ΓN ).
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The homogenized compliance is defined by
c(θ, A∗) =
∫
ΓN
g · u ds.
The relaxed or homogenized optimization problem is
min(θ,A∗)∈U∗
ad
J(θ, A∗) = c(θ, A∗) + ℓ
∫
D
θ(x) dx
.
Major inconvenient: in the elasticity setting an explicit characterization of Gθ
is still lacking !
Fortunately, for compliance one can replace Gθ by its explicit subset of
laminated composites.
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The key argument to avoid the knowledge of Gθ is that, thanks to the
complementary energy minimization, compliance can be rewritten as
c(θ, A∗) =
∫
ΓN
g · u ds = mindivσ=0 in Dσn=g on ΓN
σn=0 on ∂D\ΓN∪ΓD
∫
D
A∗−1σ · σ dx.
The shape optimization problem thus becomes a double minimization (we
already used this argument in chapter 5).
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Exchanging the order of minimizations
The shape optimization problem is
min(θ,A⋆)∈U⋆
ad
mindivσ=0 in Dσn=g on ΓN
σn=0 on ∂D\ΓN∪ΓD
∫
D
A∗−1σ · σ dx+ ℓ
∫
D
θ(x) dx
.
Since the order of minimization is irrelevent, it can be rewritten
mindivσ=0 in Dσn=g on ΓN
σn=0 on ∂D\ΓN∪ΓD
min(θ,A⋆)∈U⋆
ad
∫
D
A∗−1σ · σ dx+ ℓ
∫
D
θ(x) dx
.
The minimization with respect to the design parameters (θ, A∗) is local, thus
mindivσ=0 in Dσn=g on ΓN
σn=0 on ∂D\ΓN∪ΓD
∫
D
min0≤θ≤1A∗∈Gθ
(
A∗−1σ · σ + ℓθ)
(x) dx.
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For a given stress tensor σ, the minimization of complementary energy
minA∗∈Gθ
A∗−1σ · σ
is a classical problem in homogenization, of finding optimal bounds on the
effective properties of composite materials.
It turns out that we can restrict ourselves to sequential laminates which form
an explicit subset Lθ of Gθ.
Such a simplification is made possible because compliance is the objective
function.
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7.5.2 Sequential laminates in elasticity
Aξ = 2µAξ + λA(trξ)I, Bξ = 2µBξ + λB(trξ)I,
with the identity matrix I2, and κA,B = λA,B + 2µA,B/N . We assume B to be
weaker than A
0 ≤ µB < µA, 0 ≤ κB < κA.
We work with stresses rather than strains, thus we use inverse elasticity
tensors.
Lemma 7.24. The Hooke’s law of a simple laminate of A and B in
proportions θ and (1− θ), respectively, in the direction e, is
(1− θ)(
A∗−1 −A−1)−1
=(
B−1 −A−1)−1
+ θf cA(e)
with f cA(e) the tensor defined, for any symmetric matrix ξ, by
f cA(ei)ξ · ξ = Aξ · ξ − 1
µA
|Aξei|2 +µA + λA
µA(2µA + λA)((Aξ)ei · ei)2.
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Reiterated lamination formula
Proposition 7.25. A rank-p sequential laminate with matrix A and
inclusions B, in proportions θ and (1− θ), respectively, in the directions
(ei)1≤i≤p with parameters (mi)1≤i≤p such that 0 ≤ mi ≤ 1 and∑p
i=1 mi = 1,
is given by
(1− θ)(
A∗−1 −A−1)−1
=(
B−1 −A−1)−1
+ θ
p∑
i=1
mifcA(ei)
1ε
1εε2 >>
e2
Α =
Β =e1
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7.5.3 Hashin-Shtrikman bounds in elasticity
Theorem 7.26. Let A∗ be a homogenized elasticity tensor in Gθ which is
assumed isotropic
A∗ = 2µ∗I4 +
(
κ∗ −2µ∗
N
)
I2 ⊗ I2.
Its bulk κ∗ and shear µ∗ moduli satisfy
1− θ
κA − κ∗≤ 1
κA − κB
+θ
2µA + λA
andθ
κ∗ − κB
≤ 1
κA − κB
+1− θ
2µB + λB
1− θ
2(µA − µ∗)≤ 1
2(µA − µB)+
θ(N − 1)(κA + 2µA)
(N2 +N − 2)µA(2µA + λA)
θ
2(µ∗ − µB)≤ 1
2(µA − µB)− (1− θ)(N − 1)(κB + 2µB)
(N2 +N − 2)µB(2µB + λB).
Furthermore, the two lower bounds, as well as the two upper bounds are
simultaneously attained by a rank-p sequential laminate with p = 3 if N = 2,
and p = 6 if N = 3.
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Hashin-Shtrikman bounds in elasticity
κ
µ
θ+µ
µθ−
κ θ− κ θ
+
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Proposition 7.27. Let Gθ be the set of all homogenized elasticity tensors
obtained by mixing the two phases A and B in proportions θ and (1− θ). Let
Lθ be the subset of Gθ made of sequential laminated composites. For any
stress σ,
HS(σ) = minA∗∈Gθ
A∗−1σ · σ = minA∗∈Lθ
A∗−1σ · σ.
Furthermore, the minimum is attained by a rank-N sequential laminate with
lamination directions given by the eigendirections of σ.
Remark.
An optimal tensor A∗ can be interpreted as the most rigid composite
material in Gθ able to sustain the stress σ.
HS(σ) is called Hashin-Shtrikman optimal energy bound.
In the conductivity setting, a rank-1 laminate was enough...
Practical conclusion: Gθ can be replaced by Lθ.
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Explicit computation of the optimal bound
When B = 0 one can obtain an explicit formula for the bound:
minA∗∈Gθ
A∗−1σ · σ = HS(σ) = A−1σ · σ +1− θ
θg∗(σ)
2-D case.
g∗(σ) =κ+ µ
4µκ(|σ1|+ |σ2|)2
with σ1, σ2 the eigenvalues of σ. Furthermore, an optimal rank-2 sequential
laminate is given by the parameters
m1 =|σ2|
|σ1|+ |σ2|, m2 =
|σ1||σ1|+ |σ2|
.
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3-D simplified case with λA = 0. We label the eigenvalues of σ as
|σ1| ≤ |σ2| ≤ |σ3|.
g∗(σ) =1
4µ
(|σ1|+ |σ2|+ |σ3|)2 if |σ3| ≤ |σ1|+ |σ2|
2(
(|σ1|+ |σ2|)2 + |σ3|2)
if |σ3| ≥ |σ1|+ |σ2|
In the first regime, an optimal rank-3 sequential laminate is given by
m1 =|σ3|+ |σ2| − |σ1||σ1|+ |σ2|+ |σ3|
, m2 =|σ1| − |σ2|+ |σ3||σ1|+ |σ2|+ |σ3|
, m3 =|σ1|+ |σ2| − |σ3||σ1|+ |σ2|+ |σ3|
,
and in the second regime, an optimal rank-2 sequential laminate is
m1 =|σ2|
|σ1|+ |σ2|, m2 =
|σ1||σ1|+ |σ2|
, m3 = 0.
(General 3-D case known but more complicated.)
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7.5.4 Homogenized formulation of shape optimization
mindivσ=0 in Dσn=g on ΓN
σn=0 on ∂D\ΓN∪ΓD
∫
D
min0≤θ≤1A∗∈Gθ
(
A∗−1σ · σ + ℓθ)
dx.
Optimality condition. If (θ, A∗, σ) is a minimizer, then A∗ is a rank-N
sequential laminate aligned with σ and with explicit proportions
A∗−1 = A−1 +1− θ
θ
(
N∑
i=1
mifcA(ei)
)−1
,
and θ is given in 2-D (similar formula in 3-D)
θopt = min
(
1,
√
κ+ µ
4µκℓ(|σ1|+ |σ2|)
)
,
where σ is the solution of the homogenized equation.
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Existence theory
Original shape optimization problem
infΩ⊂D
J(Ω) =
∫
ΓN
g · u ds+ ℓ
∫
Ω
dx. (1)
Homogenized (or relaxed) formulation of the problem
minA∗∈Gθ
0≤θ≤1
J(θ, A∗) =
∫
ΓN
g · u ds+ ℓ
∫
D
θ dx. (2)
Theorem 7.30. The homogenized formulation (2) is the relaxation of the
original problem (1) in the sense where
1. there exists, at least, one optimal composite shape (θ, A∗) minimizing (2),
2. any minimizing sequence of classical shapes Ω for (1) converges, in the
sense of homogenization, to a minimizer (θ, A∗) of (2),
3. the minimal values of the original and homogenized objective functions
coincide.
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7.5.5 Numerical algorithm
Double “alternating” minimization in σ and in (θ, A∗).
• intialization of the shape (θ0, A∗0)
• iterations n ≥ 1 until convergence
– given a shape (θn−1, A∗n−1), we compute the stress σn by solving a
linear elasticity problem (by a finite element method)
– given a stress field σn, we update the new design parameters (θn, A∗n)
with the explicit optimality formula in terms of σn.
Remarks.
For compliance, the problem is self-adjoint.
Micro-macro method (local microstructure / global density).
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Remarks
The objective function always decreases.
Algorithm of the type “optimality criteria”.
Algorithme of “shape capturing” on a fixed mesh of Ω.
We replace void by a weak “ersatz” material, or we impose θ ≥ 10−3 to
get an invertible rigidity matrix.
A few tens of iterations are sufficient to converge.
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Example: optimal cantilever
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Penalization
The previous algorithm compute composite shapes instead of classical
shapes.
We thus use a penalization technique to force the density in taking values
close to 0 or 1.
Algorithm: after convergence to a composite shape, we perform a few more
iterations with a penalized density
θpen =1− cos(πθopt)
2.
If 0 < θopt < 1/2, then θpen < θopt, while, if 1/2 < θopt < 1, then θpen > θopt.
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Convergence history:
objective function (left), and residual (right),
in terms of the iteration number.
iteration number
obje
ctiv
e fu
ncti
on
0 10050
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
iteration number
conv
erge
nce
crit
erio
n
0 10050-510
-410
-310
-210
-110
010
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Example: optimal bridge
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7.5.6. Convexification and “fictitious materials”
Idea. In the homogenization method composite materials are introduced but
discarded at the end by penalization. Can we simplify the approach by
introducing merely a density θ ?
A classical shape is parametrized by χ(x) ∈ 0, 1.If we convexify this admissible set, we obtain θ(x) ∈ [0, 1].
The Hooke’s law, which was χ(x)A, becomes θ(x)A. We also call this
fictitious materials because one can not realize them by a true
homogenization process (in general). Combined with a penalization scheme,
this methode is called SIMP (Solid Isotropic Material with Penalization).
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Convexified formulation with 0 ≤ θ(x) ≤ 1
σ = θ(x)Ae(u) with e(u) = 12 (∇u+ (∇u)t) ,
divσ = 0 in D,
u = 0 on ΓD
σn = g on ΓN
σn = 0 on ∂D \ (ΓD ∪ ΓN ).
Compliance minimization
min0≤θ(x)≤1
(
c(θ) + ℓ
∫
D
θ(x)
)
.
with
c(θ) =
∫
ΓN
g · u =
∫
D
(θ(x)A)−1σ · σ = mindivτ=0 in Dτn=g on ΓN
τn=0 on ∂D\ΓN∪ΓD
∫
D
(θ(x)A)−1τ · τ dx.
Now, there is only one single design parameter: the material density θ (the
microstructure A∗ has disappeared).
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Existence of solutions
Theorem 7.33. The convexified formulation
min0≤θ(x)≤1
mindivτ=0 in Dτn=g on ΓN
τn=0 on ∂D\ΓN∪ΓD
∫
D
(θ(x)A)−1τ · τ dx+ ℓ
∫
D
θ dx
admits at least one solution.
Proof. The function, defined on IR+ ×Msn,
φ(a, σ) = a−1A−1σ · σ,
is convex because
φ(a, σ) = φ(a0, σ0) +Dφ(a0, σ0) · (a− a0, σ − σ0) + φ(a, σ − aa−10 σ0),
where the derivative Dφ is given by
Dφ(a0, σ0) · (b, τ) = − b
a20A−1σ0 · σ0 + 2a−1
0 A−1σ0 · τ.
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Optimality condition
If we exchange the minimizations in τ and in θ, we can compute the optimal θ
which is
θ(x) =
1 if A−1τ · τ ≥ ℓ√ℓ−1A−1τ · τ if A−1τ · τ ≤ ℓ
Again we can use an “alternating” double minimization algorithm.
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Numerical algorithm
• intialization of the shape θ0
• iterations k ≥ 1 until convergence
– given a shape θk−1, we compute the stress σk by solving an elasticity
problem (by a finite element method)
– given a stress field σk, we update the new material density θk with the
explicit optimality formula in terms of σk.
Penalization: we use a penalized density
θpen =1− cos(πθopt)
2or (SIMP) θpen = θp p > 1.
In practice: it is extremely simple ! But the numerical results are not as
good ! An explanation is the lack of a relaxation theorem.
Be careful: very delicate monitoring of the penalization...
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Optimal bridge by the convexification method
iteration number
com
plia
nce
0 10050
0.2
0.3
0.4
0.15
0.25
0.35
0.45
convexificationhomogenization
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Conclusion
SIMP (or convexification, or “fictitious materials”) is very simple and
very popular (many commercial codes are using it).
SIMP uses very few informations on composites !
On the contrary to the homogenization method, SIMP is not a
relaxation method: it changes the problem !
There is a gap between the true minimal value of the objective function
and that of SIMP.
SIMP can be delicate to monitor: how to increase the penalization
parameter ?
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Generalizations of the homogenization method
multiple loads
vibration eigenfrequency
general criterion of the least square type
The two first cases are self-adjoint and we have a complete understanding and
justification of the relaxation process. However, the third case is not
self-adjoint and only a partial relaxation is known.
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Multiple loads
For n loads (fi)1≤i≤n, the homogenized formulation is
mindivσi=0 in Dσin=gi on ΓN
∫
D
min0≤θ≤1
minA∗∈Lθ
(
n∑
i=1
A∗−1σi · σi + ℓθ
)
dx
with A∗ ∈ Lθ and
(1− θ)(
A∗−1 −A−1)−1
=(
B−1 −A−1)−1
+ θ
p∑
i=1
mifcA(ei)
The optimal laminate is no more of rank N . The mi’s optimization is now
done numerically (with numerous enough lamination directions).
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Optimal bridge for 3 simultaneously applied loads
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Optimal bridge for 3 independently applied loads
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Vibration eigenfrequencies
We maximize the first vibration eigenfrequency
ω21(θ, A
∗) = minu∈H
∫
D
A∗e(u) · e(u)dx∫
D
ρ|u|2dx.
with the density ρ = θρA + (1− θ)ρB , and the space of admissible
displacements H =
u ∈ H1(D)N such that u = 0 on ΓD
.
The homogenized formulation is
max0≤θ≤1
minu∈H
∫
D
(
maxA∗∈Lθ
A∗e(u) · e(u))
dx∫
D
ρ|u|2dx+ ℓ
∫
D
θ(x)dx
,
with Lθ the set of sequential laminates.
Be careful: there is a max-min which can not be exchanged...
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Least square objective functions
Classical two-phase formulation:
infχ∈L∞(Ω;0,1)
J(χ) =
∫
Ω
k(x)|uχ(x)− u0(x)|2 dx+ ℓ
∫
Ω
χ(x)dx
where uχ is solution of
− div (Aχe(uχ)) = f in Ω
uχ = 0 on ∂Ω,
with a Hooke’s law Aχ = χA+ (1− χ)B.
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Homogenized formulation:
min(θ,A∗)
J∗(θ, A∗) =
∫
Ω
(
k|u− u0|2 + ℓθ)
dx
with u solution of
− div (A∗e(u)) = f in Ω
u = 0 on ∂Ω,
Difficulty: we don’t know Gθ and we cannot replace it by Lθ. In other
words, we don’t know which microstructures are optimal...
Partial relaxation: we nevertheless replace Gθ by Lθ. We thus loose the
existence of an optimal solution but we keep the link with the original
problem.
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Partial relaxation
We restrict ourselves to sequential laminates A∗ with matrix A and inclusions
B. The number of laminations and their directions are fixed. We merely
optimize with respect to θ and the proportions (mi)1≤i≤p
(1− θ) (A−A∗)−1
= (A−B)−1 − θ
q∑
i=1
mifA(ei),
with ∀e ∈ IRN , |e| = 1, ∀ξ symmetric matrix
fA(e)ξ · ξ =1
µA
(
|ξe|2 − (ξe · e)2)
+1
λA + 2µA
(ξe · e)2.
Thus, the objective function is
J∗(θ, A∗) ≡ J∗(θ,mi)
with the constraints 0 ≤ θ ≤ 1, mi ≥ 0,∑p
i=1mi = 1.
We compute its gradient with the help of an adjoint state.
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Adjoint state
Typical example of an objective function
J∗(θ, A∗) =
∫
Ω
k(x)|u(x)− u0(x)|2dx+ ℓ
∫
Ω
θ dx
Adjoint state
− div (A∗e(p)) = 2k(x)(u(x)− u0(x)) in Ω
p = 0 on ∂Ω
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Gradient
∇θJ∗(x) = ℓ+
∂A∗
∂θe(u) · e(p),
∇miJ∗(x) =
∂A∗
∂mi
(x)e(u) · e(p),
and
∂A∗
∂θ(x) = T−1
(
(A−B)−1 −q∑
i=1
mifA(ei)
)
T−1,
∂A∗
∂mi
(x) = −θ(1− θ)T−1fA(ei)T−1,
T = (A−B)−1 − θ
q∑
i=1
mifA(ei) .
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Numerical algorithm of gradient type
Projected gradient with a variable step:
1. Initialization of the design parameters θ0,mi,0 (for example, constants
satisfying the constraints).
2. Iterations until convergence, for k ≥ 0:
(a) Computation of the state uk and the adjoint pk, with the previous
design parameters θk,mi,k.
(b) Update of the design parameters :
θk+1 = max (0,min (1, θk − tk∇θJ∗k )) ,
mi,k+1 = max (0,mi,k − tk∇miJ∗k + ℓk) ,
where ℓk is a Lagrange multiplier for the constraint∑q
i=1mi,k = 1, iteratively
updated, and tk > 0 is a descent step such that J∗(θk+1,mk+1) < J∗(θk,mk).
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Example: force inverter
C(x) = 0
F
C(x) = 1
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Other methods of topology optimization
Discrete 0/1 optimization: genetic algorithms.
Level set methods based on geometric optimization.
Topological derivative: sensitivity to the nucleation of a small hole.
Phase-field methods.
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Commercial softwares and industrial applications
See the web page:
http://www.cmap.polytechnique.fr/~optopo/links.html
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Industrial applications
Automotive industry.
Aerospace industry.
Civil engineering, architecture.
Nano-technologies, MEMS.
Optics, wave guides.
G. Allaire, Ecole Polytechnique Optimal design of structures